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Neutrino Factory/Muon Collider Document 544-v2

Twiss parameters and beam matrix formulation of generalized Courant-Snyder theory for coupled transverse dynamics and their application to helical transport channels

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Technical Note
Submitted by:
Moses Chung
Updated by:
Moses Chung
Document Created:
03 Dec 2009, 11:59
Contents Revised:
28 May 2010, 16:22
Metadata Revised:
28 May 2010, 16:22
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03 Dec 2009, 11:59
In recent papers [1, 2] Qin and Davidson have generalized Courant-Snyder (CS) theory for one
degree of freedom to the case of coupled transverse dynamics with two degrees of freedom. The
generalized theory has four basic components of the original CS theory, i.e., the envelope equation,
phase advance, transfer matrix, and the CS invariant, all of which have their counterparts in the
original CS theory with remarkably similar expressions. In this paper, we further investigate this
remarkable similarity between the original and generalized CS theory, and construct the Twiss pa-
rameters and beam matrix in generalized form, which can be used to provide a practical framework
for accelerator design, transverse beam measurement and control, and particle tracking studies.
In particular, it is shown that choosing the appropriate initial conditions for the matrix envelope
equation is important to be consistent with the symplectic condition of the transfer matrix, and
to simplify the calculation of the beam matrix. As an illustrative example, the generalized form of
the Twiss parameters and beam matrix has been applied to the case of a helical transport channel,
where the two transverse motions are strongly coupled.
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